Secant, Cosecant & Cotangent Functions | 正割、余割与余切函数

📚 Secant, Cosecant & Cotangent Functions | 正割、余割与余切函数

In A-Level Mathematics, beyond the familiar sine, cosine and tangent functions, there are three reciprocal trigonometric functions: secant (sec), cosecant (cosec or csc) and cotangent (cot). These functions appear frequently in calculus, trigonometry and exam questions involving identities, differentiation and integration. This article provides a complete revision guide to their definitions, properties, graphs, derivatives and practical problem-solving techniques.

在 A-Level 数学中，除了大家熟知的正弦、余弦和正切函数之外，还有三个倒数三角函数：正割（sec）、余割（cosec 或 csc）和余切（cot）。这些函数在微积分、三角学以及涉及恒等式、微分与积分的考试题目中频繁出现。本文提供一份完整的复习指南，涵盖其定义、性质、图像、导数及实用解题技巧。

1. Definitions | 定义

Each of the three functions is defined as the reciprocal of a primary trigonometric function, valid wherever the denominator is not zero:

这三个函数分别定义为主要三角函数的倒数，在分母不为零的所有地方均有定义：

sec x = 1 / cos x, cosec x = 1 / sin x, cot x = 1 / tan x = cos x / sin x

The notation ‘cosec’ is commonly used in UK exam boards (Edexcel, AQA, OCR), while ‘csc’ is used internationally. Both notations refer to the same function.

英国考试局（Edexcel、AQA、OCR）通常使用 ‘cosec’ 的记号，而国际课程中常用 ‘csc’。这两种记号表示同一个函数。

2. Domains and Ranges | 定义域与值域

Since sec x = 1 / cos x, it is undefined where cos x = 0, i.e. at x = π/2 + nπ. Similarly, cosec x is undefined where sin x = 0, i.e. at x = nπ. The function cot x is undefined where sin x = 0 because cot x = cos x / sin x.

由于 sec x = 1 / cos x，它在 cos x = 0 处无定义，即 x = π/2 + nπ。类似地，cosec x 在 sin x = 0 处无定义，即 x = nπ。而 cot x = cos x / sin x，所以在 sin x = 0 处也无定义。

The ranges are: sec x ≤ -1 or sec x ≥ 1; cosec x ≤ -1 or cosec x ≥ 1; cot x can take any real value.

值域方面：sec x ≤ -1 或 sec x ≥ 1；cosec x ≤ -1 或 cosec x ≥ 1；cot x 可取任意实数。

3. Key Identities | 关键恒等式

Three Pythagorean identities connect these functions to the primary ones, and they are derived directly from sin²x + cos²x = 1:

三个毕达哥拉斯恒等式将这些函数与基本三角函数联系起来，它们都直接由 sin²x + cos²x = 1 推导得出：

1 + tan²x = sec²x — divide sin²x + cos²x = 1 by cos²x
1 + cot²x = cosec²x — divide sin²x + cos²x = 1 by sin²x
tan x = sin x / cos x, cot x = cos x / sin x

These identities are essential for simplifying expressions, proving trigonometric equations and evaluating limits. In exam contexts, students often need to choose the correct identity to transform an equation into a solvable quadratic form.

这些恒等式在化简表达式、证明三角方程以及计算极限时至关重要。在考试中，学生往往需要选择正确的恒等式，将方程化为可解的二次形式。

4. Graphs of the Reciprocal Functions | 倒数函数的图像

Sketching these graphs requires an understanding of vertical asymptotes. Wherever the denominator is zero, the reciprocal function has a vertical asymptote. Between asymptotes, the graphs alternate between positive and negative branches.

绘制这些图像需要理解垂直渐近线。只要分母为零，倒数函数就存在一条垂直渐近线。在渐近线之间，图像在正、负两支之间交替变化。

For example, the graph of y = sec x has vertical asymptotes at x = π/2 + nπ, and its branches open outward from the peaks of y = cos x. The graph of y = cosec x has asymptotes at x = nπ, with branches passing through maxima and minima of y = sin x. The graph of y = cot x has asymptotes at x = nπ and crosses the x-axis at x = π/2 + nπ.

例如，y = sec x 的图像在 x = π/2 + nπ 处有垂直渐近线，其各支从 y = cos x 的峰点向外张开。y = cosec x 的图像在 x = nπ 处有渐近线，各支经过 y = sin x 的极大值与极小值点。y = cot x 的图像在 x = nπ 处有渐近线，并在 x = π/2 + nπ 处穿过 x 轴。

5. Periodicity and Symmetry | 周期性与对称性

The functions sec x and cosec x have period 2π, because they are reciprocals of sine and cosine. The function cot x has period π, matching the period of tan x.

函数 sec x 和 cosec x 的周期为 2π，因为它们是正弦与余弦的倒数。函数 cot x 的周期为 π，与 tan x 的周期一致。

Regarding symmetry: sec x is even, meaning sec(-x) = sec x, because cos x is even. Both cosec x and cot x are odd, so cosec(-x) = -cosec x and cot(-x) = -cot x. Recognizing parity is useful when evaluating definite integrals over symmetric intervals, since odd functions integrate to zero.

关于对称性：sec x 是偶函数，即 sec(-x) = sec x，因为 cos x 是偶函数。cosec x 和 cot x 都是奇函数，所以 cosec(-x) = -cosec x，cot(-x) = -cot x。识别奇偶性有助于在对称区间上计算定积分，因为奇函数在对称区间上的积分为零。

sec(-x) = sec x, cosec(-x) = -cosec x, cot(-x) = -cot x

6. Derivatives | 导数

Differentiating the reciprocal trigonometric functions is a core skill. Using the quotient rule or the chain rule from first principles, the following results hold:

对倒数三角函数求导是一项核心技能。利用商法则或链式法则，从定义出发可以证明以下结果：

d/dx (sec x) = sec x tan x

d/dx (cosec x) = -cosec x cot x

d/dx (cot x) = -cosec² x

Students should memorize these three derivatives. A common mnemonic is to remember that the derivative of each co-function (cosec and cot) carries a negative sign, whereas the derivative of sec x is positive. Applying the chain rule extends these to composite functions: for example, d/dx sec(2x) = 2 sec(2x) tan(2x).

学生应熟记这三个导数公式。一个常用的记忆方法是：余函数（cosec 和 cot）的导数都带负号，而 sec x 的导数为正。利用链式法则可将其推广至复合函数，例如 d/dx sec(2x) = 2 sec(2x) tan(2x)。

7. Integrals | 积分

The corresponding indefinite integrals are equally important. The antiderivatives of the reciprocal trigonometric functions are:

对应的不定积分同样重要。倒数三角函数的不定积分如下：

∫ sec x dx = ln|sec x + tan x| + C

∫ cosec x dx = -ln|cosec x + cot x| + C

∫ cot x dx = ln|sin x| + C

The first two integrals can also be expressed using alternative logarithmic forms, such as ln|tan(π/4 + x/2)| for sec x, but the forms above are the most practical for A-Level problems. When integrating products such as sec²x or cosec²x, recall that ∫sec²x dx = tan x + C and ∫cosec²x dx = -cot x + C.

前两个积分也可以用其他对数形式表达，例如 sec x 的积分也可写作 ln|tan(π/4 + x/2)|，但上述形式对 A-Level 题目最为实用。当积分形如 sec²x 或 cosec²x 时，记得 ∫sec²x dx = tan x + C，∫cosec²x dx = -cot x + C。

8. Solving Equations Involving Sec, Cosec and Cot | 求解含 sec、cosec 与 cot 的方程

When an equation contains sec, cosec or cot, the first step is usually to rewrite everything in terms of sin and cos, or to apply a Pythagorean identity to reduce the equation to a solvable form.

当方程中含有 sec、cosec 或 cot 时，第一步通常是把所有项改写为 sin 和 cos 的形式，或者应用毕达哥拉斯恒等式，将方程化为可解的形式。

Example: solve sec x = 2 for 0 ≤ x < 2π. Since sec x = 1/cos x, we have cos x = 1/2. Therefore x = π/3 and x = 5π/3.

例如：在 0 ≤ x < 2π 范围内解 sec x = 2。因为 sec x = 1/cos x，所以 cos x = 1/2。因此 x = π/3 和 x = 5π/3。

A more involved example: solve 2 cosec²x – cot x – 5 = 0. Using cosec²x = 1 + cot²x gives 2(1 + cot²x) – cot x – 5 = 0, hence 2cot²x – cot x – 3 = 0. Factorizing yields (2cot x – 3)(cot x + 1) = 0, so cot x = 3/2 or cot x = -1. Solutions follow by taking inverse tangent, being careful with the correct quadrants.

一个更复杂的例子：解 2 cosec²x – cot x – 5 = 0。利用 cosec²x = 1 + cot²x 得 2(1 + cot²x) – cot x – 5 = 0，即 2cot²x – cot x – 3 = 0。因式分解得 (2cot x – 3)(cot x + 1) = 0，所以 cot x = 3/2 或 cot x = -1。接着通过反正切求 x，注意选择正确的象限。

9. Inverse Functions | 反函数

Each of the three reciprocal trigonometric functions has an inverse, but domain restrictions are required to make them one-to-one. The standard principal branches are:

这三个倒数三角函数都有反函数，但需要限制定义域使其成为一一对应。标准的反正割、反余割和反余切主值分支为：

arcsec x ∈ [0, π], x ∉ (-1, 1)
arccosec x ∈ [-π/2, 0) ∪ (0, π/2]
arccot x ∈ (0, π)

These inverse functions are rarely tested in pure mathematics papers at A-Level, but they appear in some Further Mathematics syllabuses. The most common use is in integration, such as ∫ 1/(x√(x²-1)) dx = arcsec |x| + C.

这些反函数在 A-Level 纯数学试卷中很少直接考查，但在部分进阶数学大纲中会出现。最常见的用途是在积分中，例如 ∫ 1/(x√(x²-1)) dx = arcsec |x| + C。

10. Applications in Differentiation and Integration | 在微分与积分中的应用

Reciprocal trigonometric functions appear naturally when differentiating inverse trigonometric functions and when integrating rational expressions.

倒数三角函数在反三角函数求导和有理分式积分中自然出现。

For example, d/dx arctan x = 1/(1 + x²). Substituting x = tan θ gives dx = sec²θ dθ, and the expression simplifies remarkably. Similarly, integrals involving √(x² + a²) are often simplified using the substitution x = a tan θ, leading to secant terms that integrate to logarithmic functions.

例如，d/dx arctan x = 1/(1 + x²)。令 x = tan θ，则 dx = sec²θ dθ，表达式会得到极大简化。类似地，含 √(x² + a²) 的积分通常用 x = a tan θ 代换，得到含正割的项，最终积分结果含对数函数。

In the next example, evaluate ∫ sec³x dx. Multiply numerator and denominator by sec x + tan x: ∫ sec³x dx = ∫ sec x · sec²x dx = ∫ sec x d(tan x). Using integration by parts yields (1/2)(sec x tan x + ln|sec x + tan x|) + C. This technique is worth knowing for Further Mathematics examinations.

再看一个例子：计算 ∫ sec³x dx。将分子分母同乘 sec x + tan x，或使用分部积分：∫ sec³x dx = ∫ sec x · sec²x dx = ∫ sec x d(tan x)。利用分部积分得 (1/2)(sec x tan x + ln|sec x + tan x|) + C。这一技巧适用于进阶数学考试。

11. Graphs of y = a sec(bx) + c and Transformations | 函数 y = a sec(bx) + c 的图像与变换

Transformations of reciprocal trigonometric graphs follow the same rules as those for sine and cosine functions. The parameter a stretches the graph vertically, b affects the period (period = 2π/b for sec and cosec, π/b for cot), and c translates the graph vertically.

倒数三角函数的图像变换遵循与正弦、余弦函数相同的规则。参数 a 使图像纵向伸缩，b 影响周期（对 sec 和 cosec，周期 = 2π/b；对 cot，周期 = π/b），c 使图像纵向平移。

When sketching y = a sec(bx) + c, start by drawing y = a cos(bx) + c, then sketch the reciprocal branches. The stationary points of the cosine curve become the vertices of the secant branches. Always label asymptotes clearly at x-values where cos(bx) = 0.

绘制 y = a sec(bx) + c 时，先画出 y = a cos(bx) + c，再绘制倒数分支。余弦曲线的驻点成为正割分支的顶点。务必在 cos(bx) = 0 处清晰标出渐近线。

12. Exam Tips and Common Mistakes | 备考提示与常见错误

Common mistakes in exams include forgetting that sec x is undefined at cos x = 0, misapplying the Pythagorean identity (writing 1 + tan²x = cosec²x instead of sec²x), and incorrect signs in derivatives and integrals.

考试中的常见错误包括：忘记 sec x 在 cos x = 0 处无定义；错误地使用毕达哥拉斯恒等式（把 1 + tan²x = sec²x 写成 cosec²x）；以及在导数与积分中弄错符号。

Check domains: always exclude x-values where the denominator equals zero before solving equations or sketching graphs.
Remember reciprocal notation: sec x = 1/cos x, not 1/sin x.
Use identities for simplification: when an equation contains both sec²x and tan x, replace sec²x with 1 + tan²x.
Verify solutions: substitute answers back into the original equation, since squaring or multiplying may introduce extraneous roots.
Accuracy of signs in calculus: derivative of sec x is positive, while derivatives of cosec x and cot x carry negative signs.

检查定义域：在解方程或画图之前，一定要排除分母为零的 x 值。
牢记倒数关系：sec x = 1/cos x，而不是 1/sin x。
利用恒等式化简：当方程同时含 sec²x 与 tan x 时，用 1 + tan²x 替换 sec²x。
验证根：将答案代回原方程，因为平方或乘项可能产生增根。
注意微积分中的符号：sec x 的导数为正，而 cosec x 和 cot x 的导数都带负号。

Finally, practice sketching all three graphs from memory, know the standard derivatives and integrals by heart, and solve at least ten mixed problems from past papers to build confidence.

最后，请练习凭记忆绘制这三个函数的图像，熟记标准导数与积分公式，并至少完成十道来自历年真题的混合题型，以建立信心。

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Secant, Cosecant & Cotangent Functions | 正割、余割与余切函数